Lemma 2.2  [1]}

Let \(\Omega \subset \mathbb {R}^2\)  be a simply connected region and \(S \subset \Omega \)  be a regular oriented curve such that \(\partial S-\partial \Omega \subset \lbrace O\rbrace \) . Then,
the condition \(\operatorname{Curl}T=0\) , with \(T \in \mathcal {D}^{\prime }(\Omega , \mathbb {R}^2)\)  and \(T(\phi )=B(\phi )+C(\phi )\) , where \(B\in \mathcal {B}(\Omega ,\mathbb {R}^2)\) , \(C\in \mathcal {C}(\Omega ,\mathbb {R}^2)\) , and \(\phi \in \mathcal {D}(\Omega ,\mathbb {R}^2)\) , is equivalent to the existence of a scalar field \(U \in \mathcal {B}(\Omega )\)  such that \(T=\nabla U\) . If \(C=0\)  then \(U=\int _\Omega u \psi \operatorname{da}\) , for all \(\psi \in \mathcal {D}(\Omega -O)\) , such that \(u\)  is a piecewise smooth scalar field continuous across the curve \(S\) .

